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Sorry for the structure and maybe some mistakes in the post - I'm not a Mathematician and not a native English speaker, so I will use Google Translator + Grammarly for help.

Please, if you want and can - help me to find an answer to this sequence of questions:

  1. Does an inverse number to zero (null) exist? 1.1) If yes - what is it? 1.2) Or/And - is the one solution or several? 1.3) Or/And - is there an equal or different weight of these solution/-s in mathematical society?
  2. Does a meta of number exist?

My own naive thoughts (briefly and simply):

About 1 question: if we can imagine that "zero" (null) is a mathematical concept of "nothing", so maybe exists a mathematical concept of "everything"? Maybe if I call "zero" (null) an "Absolute Emptiness", so maybe we should create "Absolute Fullness"? Remark - I don't mean any sort of infinities, like infinity small values (maybe they called "infinitesimals") and their opposites - infinite big values (plus or minus, I suppose it doesn't matter in this context, and don't know their right definitions, sorry) - because these entities have values and they have some sort of scope of properties, let's call them "Relatives".

About 2 question: if we can operate not just different values, but also their "Absolute Emptiness" (which for me looks confusing, because if I have the right understanding, we trying to operate by "nothing" with serious faces =))) - maybe there is an existing core property of a number, lets call it's "meta". This "meta" means the existence of some object, which we want to operate. Or maybe, it would be useful to separate operating of "Absolutes" (like only two objects - "zero" (null) and it's inverse) and "Relatives" which has a range from infinity small number (no matter of sign) to infinity big number (also not matter of sign)? If we do this separation, maybe we displace problems like division by zero or just will look at the problem from another angle - because it seems like when we dividing the non-zero value by zero (or do other 3 arithmetic operations) we work with 2 types of entities/properties ("Relative" and "Absolute") which maybe require a different approach, because for my understanding (of course with Dunning–Kruger effect xDD) there is no fundamental/crucial/disbalance problem to divide non-zero infinity small/big/mediocre value by non-zero infinity small/big/mediocre value (or do other 3 arithmetic operations, or other general operations), are you agree?

It would be so nice if you give me answers to those questions because I even don't how trivial these questions are.

P.S. If there are no one/multiple clear answers, maybe you can suggest a path/sequence of books (from zero to goal) which I must read/understand and that is will help me solo find answers to the questions above.

Thanks you in advance.

  • See: https://math.stackexchange.com/questions/377042/allowing-the-zero-element-in-a-field-to-have-an-inverse and https://math.stackexchange.com/questions/833132/multiplicative-inverse-of-0 – Adam Rubinson Nov 23 '20 at 12:07
  • @AdamRubinson thank you for the response, I need more time to fully understand info from links –  Nov 23 '20 at 12:36

1 Answers1

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Assume there was some number $\star$ that we add to the real numbers $\mathbb R$ with the property $0\cdot \star = \star \cdot 0 = 1$. From this it follows that for any number $x\in\mathbb R$ we have $$ x = x\cdot 1 = x\cdot 0\cdot \star = 0\cdot \star =1. $$ This is a contradiction of course, since there are real numbers that are not equal to $1$.

Hence, you can't add a multiplicative inverse of $0$ to the reals. If you attempt to, you end up with a structure where all elements are equal to $1$, the zero ring.

Christoph
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  • Thank you too for the response, but your explanation is slightly confusing me, if we look at the [explanation] (https://math.stackexchange.com/questions/1009868/why-does-zero-raised-to-the-power-of-negative-one-equal-infinity) we can see, that quantity of zero raised to the power of negative one is indefinite. Maybe the property of zero and his inversion has some extension outside of real numbers? –  Nov 23 '20 at 12:48
  • @IhorIvliev This is just a symbol, feel free to replace "$0^{-1}$" in my answer by any other symbol you like. Maybe $\star$. The argument doesn't depend on how we call this potential additional number. You know what, I'll do the replacement! – Christoph Nov 23 '20 at 13:00
  • Alright) thank you very much for your time and effort, especially for the ⋆ ;D I believe that at my current stage of development, your answer is perceived as a laconic, correct, and beautiful solution. My thoughts finally can focus on more applied work tasks) –  Nov 23 '20 at 13:25